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\title{ \ifdemonstrator
\textbf{DEMONSTRATOR NOTES}
\\
\fi
School of Physics and Astronomy\\
	\textbf{ASTRONOMY 2 LABORATORY}\\
	Special Relativity}
\date{} % Activate to display a given date or no date (if empty),
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\begin{document}
\maketitle
\textit{Recommendation: students are encouraged to do this lab after all special relativity lectures.}
\section*{\textbf{Purposes}}
To develop an understanding of some special relativity concepts and the effects of near light speed travel. Also, peform computer simulation of the experiment with muons to show time dilation.
\section*{\textbf{Task List}}

\begin{enumerate} \itemsep1mm \parskip-1pt
\item {Understand special relativity and main ideas behind it.}
\item {Identify the headlight effect, Doppler shift, light delay, aberration, distortion of objects, relativity of simultaneity, Terrel rotation.}
\item {Demonstrate Lorentz Transformations using RTR.}
\item {Know the difference between rest and relativistic energies and their relation with velocity.}
\item {Show time dilation in muon decay and find $\gamma$ factor.}
\end{enumerate}

\section*{\textbf{Summary}}
The revolutionary paper “On the electrodynamics of moving bodies" was published by Einstein in 1905. It proposed such ideas as time being observed to run slower for a moving body than for a stationary one or that lengths are observed to be contracted from the observer's position as you increase your speed. These and other surprising effects are the results of only two axioms, introduced by Einstein:

\begin{itemize}
\item \textit{All inertial frames are equivalent for the performance of all physical experiments.}
\item \textit{The speed of light, c, is a universal constant, the same in any inertial frame.}
\end{itemize}

Even now, more than 100 years from the publication, some people find special relativity theory counterintuitive since it isn't obvious in everyday life where speeds are too small to see the effects of SR. 
\\
\\
To begin the experiment, you will be asked to analyze different reference frames and  investigate the headlight effect, Doppler shift, distortion of objects, light delay and aberration. You will have to verify Lorentz transformations for length and time by taking measurements within the program called Real Time Relativity.
\\
\\
In the second part of the experiment you will be asked to prove time dilation for muons - particles which formed at the atmosphere and were detected. This can be done by measuring count number at different heights, $h_1$ and $h_2$, finding muon's life-time and its speed.  


\section*{\textbf{Part one}}

\subsection*{Background}
Lorentz Transformations represents the implications of Einstein's two postulates in the real world. Say that the object moves at speed v relatively to the observer, then the time goes slower for the object than for the observer. It's called time dilation and can be expressed as:
\begin{equation*}\Delta t = \Delta t_0 \gamma, \end{equation*} where $\Delta t_0$  is the time between two events measured by an observer who is present at both events, $\Delta t$ is the time between events for the moving object. $\gamma = \frac{1}{\sqrt{1-v^2/c^2}},$ where $v$ is relative speed, $c$ - the speed of light.
\\
Moreover, there is a length contraction as well which is given by the formula:
\begin{equation*}\Delta L = \frac{\Delta L_0}{\gamma}, \end{equation*} where $\Delta L$ is the length of the object as measured by a co-moving observer, $\Delta L_0$ is the length of the object when it's stationary and $\gamma$ is the same as before.
\\
\\
You will use \textbf{the Real Time Relativity (RTR)} program to perfom this part of the experiment. \textbf{RTR} is a 3D virtual reality simulation that is able to show what you would see if you were travelling at the speeds near the speed of light. In order to make it, objects scale had to be increased to enormous sizes - for example a cubed shaped clock in the scenario \textbf{Relativity of Simultaneity} has the length of 1 light second or equal to 300 000km. 
\\
When you start \textbf{RTR} you enter to the Main Menu, where you can see options:``Scenarios", ``Controls" and ``Exit". To return to this screen at any time, press the ``Escape" key (ESC). 

\subsection{Familiarising yourself with Real Time Relativity}
Enter the \textbf{Main menu}, go to \textbf{Scenarios} and double-click on \textbf{Relativity of simultaneity}.
\\
You can see the rocket which you can control with the mouse. Hold the left mouse button down while moving the mouse to steer the rocket.  The \textbf{W} and \textbf{S} keys control acceleration and deceleration (\textit{Suggestion: try to increase rocket acceleration slowly and have a look at the speed in the on-screen display as you accelerate to understand how the effects changes with speed. It's hard to find out what's going on from the first time. Useful key - \textbf{M}  - turns off relativistic effects}). Anytime you can press \textbf{F2} to pause the simulation. Then you are able to change your view direction, or inertial frame, while staying at the same event. Go to \textbf{Controls} to investigate which keys are responsible for changing the view direction, reseting the configuration, etc. Useful keys to remember: \textbf{=}, \textbf{F2}, \textbf{L}, \textbf{0} (there are more!). 
\\
\textit{Note that at left of the RTR screen are a number of clickable icons. Clickable icons are responsible for (from bottom to top): clock zeroing, clock overlay, pausing, the headlight effect and the Doppler effect.
\\
Try to use scenario \textbf{Landscape} for the first time in \textbf{RTR}}.
\\
You should spend at least 15-20 minutes for investigating the setup.
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: Although main controls cannot be changed, steering wheel and display option can be changed (Main menu $\rightarrow$ Options). If there is something going wrong, check these settings and change if required.}
\fi
\subsection{The effects of special relativity}
The view of our world at high speeds dramatically changes due to effects of SR. Run the simulation named \textbf{ Relativity of simultaneity} and identify each of the following effects. You should take screen printouts of each effect with explaining what you see and what caused it (if it's not explaned here).

\begin{enumerate} \itemsep1mm \parskip-1pt
\item {headlight effect}
\item {Doppler shift (\textit {have you solved an example from 4 section of SR notes?})}
\item light delay (\textit{remember that light travels in finite speed})
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: the easiest way to see light delay is to stop the ship near the clocks in the scenario Simultaneity of relativity and look at different clocks at one time (i.e. press O $\rightarrow$ Observe stationary)}
\fi

\item {aberration - it's the difference in the angle of incoming light rays measured by relatively moving observers. Pretty simple analogue can be drawn from the classical description of aberration which is illustrated below.
\begin{figure}[h!]
  \caption{Classical aberration where $v_{rain} = const$ and $v_{train}$ is showed under the images.
\\
Source: http://www.fourmilab.ch/cship/aberration.html}
  \centering
    \includegraphics[width=0.6\textwidth]{aberration.jpg}
\end{figure}
}
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: aberration can be clearly seen at scenario Landscape or Cube lattice.}
\fi
\item {Terrel rotation - moving object appear rotated; this is a combined effect of aberration and length contraction.
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: scenario Simultaneity of relativity, press L (to look at left) and watch rotated clocks.}
\fi}
\item {distortion of objects -  go to scenario Landscape, press O $\rightarrow$ Skyscrapers and increase your speed. What happens to skyscrapers? What causes it? You may find useful the image below.
\begin{figure}[h!]
  \caption{Distortion of rectangle.
\\
Source: http://www.anu.edu.au/Physics/vrproject/rtr.html}
  \centering
    \includegraphics[width=0.6\textwidth]{distortion2.jpg}
\end{figure}
}
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: skyscrapers look bent over because of the finite speed of light (light delay) and aberration. }
\fi
\item {relativity of simultaneity}
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: the key idea here is to create conditions such that light delay could be neglected. In order to do that, firstly go to scenario ``Relativity of simultaneity'' $\rightarrow$ O $\rightarrow$ Observe stationary, have a look how watches are synchronized and then go O $\rightarrow$ Observe moving and look at synchroniation.}
\fi
 \end{enumerate}
\ifdemonstrator
\textbf{DEMONSTRATOR: The purpose of first part of the lab is to make students ask questions, have a discussion and attempt to identify problems and answers by themselves. Try not to give them answers or hints but to raise a discussion. Example: in the end of first part you could ask ``How the world would change if the light speed would be infinite?'' }
\fi
\subsection{Demonstrating length contraction and time dilation}
In this step you will perform your own study of RTR virtual world to illustrate Lorentz Transformations for length and time. The panel on the top left of the RTR screen shows coordinates, times and other info which you could find useful.  Think about relativistic effects and how they effect the view of world from your viewpoint.
\\
Describe your experiment, noting the specifics of what you do. Provide tables of your data and the analysis required to verify length contraction and time dilation.
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: data should be gathered at constant speeds (use keys = or -) and looking at the top left panel for coordinates. Coordinates can be chosen by pausing the game (key F2) and using mouse to change the reference frame (i.e. coordinates). Record world and proper times, and how your coordinates changes by time. Note that this is only one of possible solutions for this problem. Students should be encouraged to record the data for different speeds (there are two speeds: $\gamma = 2 = 0.866c$ and $\gamma = 4 = 0.968c$).}
\fi

\section{\textbf{Part two}}
\subsection*{Background}
Now you will be retracing one of experiments made to confirm time dilation as predicted by the special relativity. Highly energetic cosmic rays enters Earth atmosphere and about 20km altitude new particles - muons - are created. They travel near the speed of light and some of them can be detected with a scintillator. They have life-time, $\tau$, and decay into other particles in a way which is described by the exponential decay law:
\begin{equation*}N = N_0 e^{-T/\tau},\end{equation*}
where $N$ is the number of undecayed particles, $N_0$ is the number of sample particles and T is decay process time.
\\
The number of decayed (and detected) muons depends only on the travel time which can be calculated from data measures at $h_1$ and $h_2$. However, there is a big difference between theoretical and measured numbers of muons detected which can be explained only by the time dilation. In other words, the discrepancy is caused by different frames and different time flow rates.

\subsection{Taking data using \textbf{Muon simulation.exe}}
\textit{Muon simulation reproduces the muon detection at the altitudes from 0 to 3000m and recording time from 1 to 10 hours. Note that not all muons are caught in the scintillator. Iron slabs are put on the detector to record muons and their life-time with specific energy E. Muons which have smaller energy than E doesn't succeed going through slabs, while muons with higher energy goes straight through the slabs and detector.}
\\
Run the simulation for different altitudes and note down the number of muons counted, altitude and recording time. Does the exposure time play any part in data collection?
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: More collection time gives more accurate count number since it's Poisson distribution.}
\fi
\subsection{Finding mean life-time}
Use the simulation again and find the mean life-time for muon. Show the plot with error estimations. Which muons' life-time is measured in the experiment - rest or (and?) moving ones? Why?
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: Life-time is measured for muons at rest since they have left only rest energy after passing the slabs.}
\fi
\subsection{Muons travel speed}
It's given that on Mt. Washington (1909 meters) scientists used $600g/cm^2$ of iron. Calculate the speed of muons which were detected by a scintillator. What area density of iron slabs need to be used at sea level for same muons?  See the Appendix for additional information. 
\\
One of assumptions for this simulation is that muons neither created nor decayed between measuring heights. Find $\Delta h$ at which this assumption brokes down.
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: Speed can be found by the relation $E = \gamma m_0 c^2$ and taking the value for E from the Appendix. To find area density: find area density for air ($13.5 * (76 - 60) = 217.6g/cm^2$) and subtract from area density of iron: ($600 - 217.6 = 382.4g/cm^2$).
\\
$\Delta h$ can be found by taking full decay time, i.e. $2.2\mu s$, and multiplying by muon travel speed.}
\fi
\subsection{Finding $\gamma$ factor}
Using your collected data from the simulation and travel speed find the $\gamma$ factor and plot the graph of muon counts clearly marking counts with relativistic time dilation and without. Discuss $\gamma$ factor relation to speed (\textit{i.e.how speed changes from $\gamma = 5$ to $\gamma = 50$?}).
\\
\textit{Hint: $\gamma$ factor can be found by comparing $\Delta t'$ and $\Delta t$.} 
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: $\Delta t$ can be found simply by $\Delta h / v_{muon}$, while $\Delta t'$ value we get from calculations using explonential decay law. This particular lab part is useful for student to distinguish one reference frame from another and understand that although measurements are different at different ref. frames, they see the same result (same count number).}
\fi
\subsection{Assumptions}

All experiments are done with some assumptions because of apparatus imperfection, lack of theoretical knowledge, etc. Discuss with your lab friend or demonstrator what assumptions are used for this experiment. How they can be corrected?
\ifdemonstrator
\\
\textbf{DEMONSTRATOR: most of the assumptions: Poissonian source; no interactions during flight; all counted and measured muons travel at the same speed; no accidental decays of other particles; effects of non-vertical muons is negligible; no latitude difference between measuring places; atmosphere density is constant across measuring altitude.}
\fi
\\
\\
\\
\textit{This experiment is based on the article``Measurement of the Relativistic Time Dilation using $\mu$-Mesons" by Frisch and Smith.}
\ifdemonstrator
\\
\\
\textbf{DEMONSTRATOR: This lab has been introduced during 2013-2014 session and could be changed on behalf of students and demonstrators responses. There has been made a request for students who are doing this lab to answer the questionnaire and put it to the lab script. Please collect all completed questionnaires and give them for lab head Dr. Ik Siong Heng.
\\
\\
If you have any suggestions or questions about this lab, please contact by email one of the following: Norman Gray (norman@astro.gla.ac.uk), Ik Siong Heng (Ik.Heng@glasgow.ac.uk) or Ronaldas Macas (1103799M@student.gla.ac.uk).}
\fi

\section*{\textbf{Appendix}}
\begin{figure}[h!]
  \centering
    \includegraphics[width=0.8\textwidth]{muon_total_energy.jpg}
\end{figure}

\begin{table}[h]
 \centering
\begin{tabular}{|r|l|}
  \hline
  Muon mass & $106 MeV/c^2$ \\
  \hline
 $P_{Mt.\, Washington}$ & 60 cm Hg \\
  \hline
  $P_{sea\: level}$ & 76 cm Hg \\
  \hline
  $\sigma_{mercury}$  & $13.6 g/cm^3$ \\
  \hline
\end{tabular}
\end{table}

\newpage

\ifdemonstrator
\else

\section*{\textbf{Questionnaire}}
This lab has been introduced during 2013-14 session so your evaluation could help us to make it better. Please answer questions below and add the questionnaire to your lab script.
\\
 \begin{enumerate} \itemsep1mm \parskip-1pt
\item {This lab helped me to understand SR better.
\\
\\
\mtab{agree}  \btab{don't agree}  \btab{don't know}
\\}
\item {I would encourage other students to do it.
\\
\\
\mtab{agree}  \btab{don't agree}  \btab{don't know}
\\
}
\item Do you think the lab was well-designed (1 -- bad, 5 -- good)?
\\
\\
 \itab{1}  \tab{2}  \tab{3}  \tab{4}  \tab{5}
\\
\item If there was any part of the lab where it was hard to understand what you
needed to do, write the section number below.
\\
\\
\item {Any more comments?}
\end{enumerate}
\fi

\end{document}
